(i) if f(x) is an odd function, then ∫f(x) x ∈[-a,a] =? (a) 0 (b) 1

Question

(i) If f (x) is an odd function, then\(\int\limits_{-a}^af(x)\) = ?
(a) 0
(b) 1
(c) 2\(\int\limits_0^a\)f(x) dx
(d) 2a

Evaluate

(ii)\(\int\limits_{-\frac{π}{2}}^{\frac{π}{2}} sin^{99}.cos^{100}xdx\)

(iii) \(\int\limits_{-1}^1 e^{|x|} dx\)

Answer 1

(i) (a) 0.

(ii) Here, f(x) = sin⁹⁹x.cos¹⁰⁰x .then,

f(-x) = sin⁹⁹(- x).cos¹⁰⁰(- x) = – sin⁹⁹ x. cos¹⁰⁰ x = -f(x)

∴ odd function ⇒

\(\int\limits_{-\frac{π}{2}}^{\frac{π}{2}} sin^{99}.cos^{100}xdx\) =0.

(iii) Here, f(x) = e^|x|, f(-x) = e^|-x| = e^|x| = f(x)

∴ even function.

we have |x| = x, 0 ≤ x ≤ 1